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where ln denotes the natural logarithm, is the thermodynamic equilibrium constant, and R is the ideal gas constant.This equation is exact at any one temperature and all pressures, derived from the requirement that the Gibbs free energy of reaction be stationary in a state of chemical equilibrium.
The equilibrium constant of a chemical reaction is the value of its reaction quotient at chemical equilibrium, a state approached by a dynamic chemical system after sufficient time has elapsed at which its composition has no measurable tendency towards further change.
The following list includes the continued fractions of some constants and is sorted by their representations. Continued fractions with more than 20 known terms have been truncated, with an ellipsis to show that they continue. Rational numbers have two continued fractions; the version in this list is the shorter one.
In differential geometry, a constant scalar curvature Kähler metric (cscK metric) is a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is a Kähler–Einstein metric , and a more general case is an extremal Kähler metric .
A sphere of radius R has constant Gaussian curvature which is equal to 1/R 2. At the same time, a plane has zero Gaussian curvature. At the same time, a plane has zero Gaussian curvature. As a corollary of Theorema Egregium, a piece of paper cannot be bent onto a sphere without crumpling.
The theorem asserts the existence of positive constants C, K such that given any n points in the plane, at least one of the following statements is true: There is a line which contains at least n/C of the points. There exist at least / lines, each of which contains at least two of the points.
A Kähler manifold is a Riemannian manifold of even dimension whose holonomy group is contained in the unitary group (). [3] Equivalently, there is a complex structure on the tangent space of at each point (that is, a real linear map from to itself with =) such that preserves the metric (meaning that (,) = (,)) and is preserved by parallel transport.
The derivative of a constant term is 0, so when a term containing a constant term is differentiated, the constant term vanishes, regardless of its value. Therefore the antiderivative is only determined up to an unknown constant term, which is called "the constant of integration" and added in symbolic form (usually denoted as ).